\chapter{Riemann zeta function and related}


Strong links to analytic number theory.

\section{Riemann zeta function}

\begin{definition}[Riemann zeta function]
\[\zeta(s) = \sum^{\infty}_{n=1} \frac{1}{n^s}\]
\end{definition}

\subsection{Euler's product formula}
The following property due to Euler connects this function intimately to number theory.


\section{Riemann xi function}


\begin{definition}[Riemann xi function]
\[\xi (s) = \frac{s(s-1)}{2} \pi^{-s /2} \Gamma ( \frac{s}{2}) \zeta (s)\]
\end{definition}